SDE Statistics Mini Course Topics List for the

SDE & Statistics Mini. Course Topics List for the Exam Università di Verona, 2015 may 25 Michele Bonollo michele. bonollo@imtlucca. it

Topics (until may 25 th) • Stochastic processes • The Ito Formula • Meaning of the second order term • Basic Properties of the Brownian motion • Black&Scholes Model • The context • Parameters practical meaning • The SDE and the log normal solution • ITO proof of the solution • Options pricing • Call & Put options • Pay. Off concept • Fair value & mark to market concept. B&S formula. Call-Put Parity • Greeks. Delta, Gamma, Vega for the call option • Montecarlo Pricing and the variance estimation (inf & SUP)

Topics (until may 25 th) • Option Risk measurement • The Leverage of an Option and Portfolio of Options • The general Va. R (quantile) definition • The Delta Va. R vs the “full evaluation” Va. R for one options • Portfolio volatility and Delta. Va. R of an options portfolio • Practical Applications (select 1 of them or a mixed recipe) • The Brownian motion and lognormal simulation. Exercise: To generate for a lognormal diffusion St at different time steps ti N paths and to estimate at any time step the average, the standard deviation and the quantile (alfa level) of St. Improvement: to make it for two diffusions S 1 & S 2 with a given correlation r • Montecarlo estimation of the fair value & Montecarlo volatility. Upper and lower bound for the price (portfolio of options). Exercise: improve what we did in the course. Improvement: Calculate the Mt. M of a portfolio of 2 options with the same underlying diffusion St. • What If analysis. Greeks accuracy vs. Full evaluation approach. Exercise: as in the course. Improve it with a grid of M different scenarios at the same time and try to plot in a graph the dependence of the Mt. M from the Underlying price. Scenarios could be ± 1% ± 2%, … ± 10% in the underlying level St • Delta impact in practical cases. Exercise: as in the course calculate the Profits & Loss effect with the delta approximation and the Va. R of the option. Estimate the volatility s by an actual stock price time series, choice an index (MIB, DAX, S&P 500) or a share (UNICREDIT, TELECOM, FINMECCANICA, . . ) as you prefer